Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control

Analytical Expressions of the Dark Resonance Parameters in a Vacuum Vapor Cell

Denis V. Brazhnikov, Grégoire Coget, Moustafa Abdel Hafiz, Vincent N. Maurice, Christophe Gorecki, Rodolphe Boudot

Published Mar 1, 2018• 2 citations
Publisher

Abstract

This paper reports a dedicated theoretical and experimental study on the properties (signal amplitude and linewidth) of coherent population trapping resonances detected in vacuum vapor cells. Results are presented for conventional single-lambda schemes of atomic energy levels but also for double-lambda schemes, now widely used in various applications including atomic clocks and magnetometers. Approximate compact analytical expressions, valid for a wide range of light-wave intensities, i.e., beyond the low intensities or pump-probe regime, have been obtained. Analytical results are found to be in excellent agreement with exact numerical solutions based on the optical Bloch equations. Experimental results, obtained in a Cs vapor microfabricated cell, are reported and found to be in correct agreement with theoretical expressions.

DOI
10.1109/tuffc.2018.2811319

Related papers

Analytical expressions for parameters of the dark resonance in a vacuum vapour cell

Denis V. Brazhnikov, Moustafa Abdel Hafiz, Grégoire Coget +1 more

In spite of great number of papers devoted to theoretical and experimental study of the coherent population trapping phenomenon (CPT), still there is a lack of analytical results for the properties of the CPT-caused dark resonance in some principal cases. We will present the results of our study for single and double A schemes of atomic energy levels, which are of great importance for various applications of CPT (atom clocks, magnetometers, etc.). Analytical expressions for width and amplitude of the dark resonances have been obtained, which are valid for wide range of light wave intensities. The results concern closed as well as open A schemes. We will also discuss the difference between the two A schemes and state the condition when a double scheme can be effectively treated as a single A scheme. The analytical approximate formulas obtained have been compared with exact numerical solutions based on the optical Bloch equations as well as experimental.

Joint Conference of the European Frequency and Time Forum and IEEE International Frequency Control Symposium (EFTF/IFCS)•Jul 1, 2017•0 citations

Coherent-population-trapping resonances in buffer-gas-filled Cs-vapor cells with push-pull optical pumping

Xiaochi Liu, Jean-Marc Mérolla, Stéphane Guérandel +3 more

We report on a theoretical study and experimental characterization of coherent-population-trapping (CPT) resonances in buffer-gas-filled vapor cells with push-pull optical pumping (PPOP) on Cs ${D}_{1}$ line. We point out that the push-pull interaction scheme is identical to the so-called $\mathrm{lin}\ensuremath{\perp}\mathrm{lin}$ polarization scheme. Expressions of the relevant dark states, as well as of absorption, are reported. The experimental setup is based on the combination of a distributed feedback diode laser, a pigtailed intensity Mach-Zehnder electro-optic modulator (MZ EOM) for optical sideband generation and a Michelson-like interferometer. A microwave technique to stabilize the transfer function operating point of the MZ EOM is implemented for proper operation. A CPT resonance contrast as high as 78$%$ is reported in a cm-scale cell for the magnetic-field-insensitive clock transition. The impact of the laser intensity on the CPT-clock signal key parameters (linewidth, contrast, linewidth/contrast ratio) is reported for three different cells with various dimensions and buffer-gas contents. The potential of the PPOP technique for the development of high-performance atomic vapor-cell clocks is discussed.

arXiv:1302.3724•Physical Review A•Jan 17, 2013•77 citations

Coherent Population Trapping with High Common-Mode Noise Rejection Using Differential Detection of Simultaneous Dark and Bright Resonances

Peter Yun, Rodolphe Boudot, Emeric de Clercq

Coherent manipulation of atomic states is highly desirable in numerous applications spanning from fundamental physics to metrology. In this study, we propose and demonstrate, theoretically and experimentally, the simultaneous observation in a vapor cell of dark and bright resonances that provide, through a differential-detection stage, an output coherent-population-trapping atomic resonance that benefits from a doubled amplitude and high common-mode noise rejection. This advanced spectroscopic scheme might be of interest for the development of high-performance vapor-cell atomic clocks, sensors, or high-resolution spectroscopy experiments.

Physical Review Applied•Feb 3, 2023•15 citations

Derivation of coherent-population-trapping resonance signals from density-matrix equations with all relevant sublevels of Cs atoms and confirmation of experimental results

Kenta Matsumoto, Sota Kagami, Takahiro Fujisaku +4 more

Coherent-population-trapping resonance in vapor cell is a quantum interference effect that appears in the two-photon transitions between the ground-state hyperfine levels of alkali-metal atoms and is often utilized in miniature and centimeter-scale clock devices. To quantitatively understand and predict the performance of this phenomenon, it is necessary to consider the transitions and relaxations between all hyperfine Zeeman sublevels involved in the different excitation processes of the atom. In this paper, we constructed a computational multilevel atomic model of the Liouville density-matrix equation for 32 Zeeman sublevels involved in the ${D}_{1}$ line of $^{133}\mathrm{Cs}$ irradiated by two frequencies with circularly polarized components and then simulated the amplitude and shape of the resonance spectrum of the transmitted light through centimeter-scale Cs-vapor cells. We show that the numerical solutions of the equation and analytical investigations adequately explain a variety of the characteristics observed in the experiment.

arXiv:2309.06761•Physical Review A•Aug 27, 2024•1 citation

Characterization of coherent population-trapping resonances as atomic frequency references

Svenja A. Knappe, R. Wynands, John E. Kitching +2 more

A low-cost, potentially compact and robust microwave frequency reference can be constructed by use of vertical-cavity surface-emitting lasers and coherent population-trapping resonances in Cs vapor cells. Fractional frequency instabilities of 2×10-11/τ/s have been achieved with a minimum of 7×10-13 at τ=1000 s. The performance of this device as a function of external parameters such as light intensity, optical detuning, and cell temperature is discussed. The dependence of the dark-line resonance signal on these parameters can be understood largely by means of a simple, three-level model. The short-term stability depends critically on the optical detuning, whereas the long-term stability is limited currently by line shifts due to drifts in cell temperature.

Journal of the Optical Society of America B•Nov 1, 2001•144 citations

Light shift mitigation in microcell-based coherent-population-trapping atomic clocks in the field of two circularly polarized light beams

Denis V. Brazhnikov, S M Ignatovich, Mikhail N Skvortsov

State-of-the-art miniature atomic clocks (MACs) are based on the phenomenon of coherent population trapping (CPT) in alkali-metal atomic vapors ($\mathrm{Rb}$ or $\mathrm{Cs}$). Improving the stability of the clock frequency is a crucial issue; resolving it would result in significant progress in numerous applications. Here, we examine the configuration of the light field, which is composed of two bichromatic beams with opposing handedness of circular polarization. The beams are in resonance with optical transitions in the $\mathrm{Cs}$ ${D}_{1}$ line ($\ensuremath{\lambda}\ensuremath{\approx}895$ nm). This configuration has previously been shown to produce observed CPT resonances with higher contrast than a standard single-beam scheme; however, our scheme implements pump and probe beams that operate independently, with the transmission of the probe beam being monitored separately. This approach differs from previous studies. The experiments are carried out using a cubic $5\ifmmode\times\else\texttimes\fi{}5\ifmmode\times\else\texttimes\fi{}5\phantom{\rule{0.2em}{0ex}}{\mathrm{mm}}^{3}$ vapor cell filled with buffer gas. It is shown that the line shape of the resonance acquires an asymmetry that can be efficiently controlled by a microwave (Raman) phase between the beams. As a proof of concept, we study the way in which this asymmetry can help to mitigate the influence of the ac Stark (light) shift on the long-term frequency stability of CPT clocks based on vapor microcells ($V\ensuremath{\ll}1\phantom{\rule{0.2em}{0ex}}{\mathrm{cm}}^{3}$). Experimental verification is performed with both a distributed-Bragg-reflector laser and a vertical-cavity surface-emitting laser. The latter is of particular importance for the development of MACs. The experimental results are in qualitative agreement with the analytical theory based on a double-$\mathrm{\ensuremath{\Lambda}}$ scheme of atomic energy levels.

arXiv:2311.00461•Physical Review Applied•May 23, 2024•8 citations
This paper reports a dedicated theoretical and experimental study on the properties (signal amplitude and linewidth) of coherent population trapping resonances detected in vacuum vapor cells. Results are presented for conventional single-lambda schemes of atomic energy levels but also for double-lambda schemes, now widely used in various applications including atomic clocks and magnetometers. Approximate compact analytical expressions, valid for a wide range of light-wave intensities, i.e., beyond the low intensities or pump-probe regime, have been obtained. Analytical results are found to be in excellent agreement with exact numerical solutions based on the optical Bloch equations. Experimental results, obtained in a Cs vapor microfabricated cell, are reported and found to be in correct agreement with theoretical expressions.
In spite of great number of papers devoted to theoretical and experimental study of the coherent population trapping phenomenon (CPT), still there is a lack of analytical results for the properties of the CPT-caused dark resonance in some principal cases. We will present the results of our study for single and double A schemes of atomic energy levels, which are of great importance for various applications of CPT (atom clocks, magnetometers, etc.). Analytical expressions for width and amplitude of the dark resonances have been obtained, which are valid for wide range of light wave intensities. The results concern closed as well as open A schemes. We will also discuss the difference between the two A schemes and state the condition when a double scheme can be effectively treated as a single A scheme. The analytical approximate formulas obtained have been compared with exact numerical solutions based on the optical Bloch equations as well as experimental.
We report on a theoretical study and experimental characterization of coherent-population-trapping (CPT) resonances in buffer-gas-filled vapor cells with push-pull optical pumping (PPOP) on Cs D1{D}_{1}D1​ line. We point out that the push-pull interaction scheme is identical to the so-called $\mathrm{lin}\ensuremath{\perp}\mathrm{lin}$ polarization scheme. Expressions of the relevant dark states, as well as of absorption, are reported. The experimental setup is based on the combination of a distributed feedback diode laser, a pigtailed intensity Mach-Zehnder electro-optic modulator (MZ EOM) for optical sideband generation and a Michelson-like interferometer. A microwave technique to stabilize the transfer function operating point of the MZ EOM is implemented for proper operation. A CPT resonance contrast as high as 78% is reported in a cm-scale cell for the magnetic-field-insensitive clock transition. The impact of the laser intensity on the CPT-clock signal key parameters (linewidth, contrast, linewidth/contrast ratio) is reported for three different cells with various dimensions and buffer-gas contents. The potential of the PPOP technique for the development of high-performance atomic vapor-cell clocks is discussed.
Coherent manipulation of atomic states is highly desirable in numerous applications spanning from fundamental physics to metrology. In this study, we propose and demonstrate, theoretically and experimentally, the simultaneous observation in a vapor cell of dark and bright resonances that provide, through a differential-detection stage, an output coherent-population-trapping atomic resonance that benefits from a doubled amplitude and high common-mode noise rejection. This advanced spectroscopic scheme might be of interest for the development of high-performance vapor-cell atomic clocks, sensors, or high-resolution spectroscopy experiments.
Coherent-population-trapping resonance in vapor cell is a quantum interference effect that appears in the two-photon transitions between the ground-state hyperfine levels of alkali-metal atoms and is often utilized in miniature and centimeter-scale clock devices. To quantitatively understand and predict the performance of this phenomenon, it is necessary to consider the transitions and relaxations between all hyperfine Zeeman sublevels involved in the different excitation processes of the atom. In this paper, we constructed a computational multilevel atomic model of the Liouville density-matrix equation for 32 Zeeman sublevels involved in the D1{D}_{1}D1​ line of 133Cs^{133}\mathrm{Cs}133Cs irradiated by two frequencies with circularly polarized components and then simulated the amplitude and shape of the resonance spectrum of the transmitted light through centimeter-scale Cs-vapor cells. We show that the numerical solutions of the equation and analytical investigations adequately explain a variety of the characteristics observed in the experiment.
A low-cost, potentially compact and robust microwave frequency reference can be constructed by use of vertical-cavity surface-emitting lasers and coherent population-trapping resonances in Cs vapor cells. Fractional frequency instabilities of 2×10-11/τ/s have been achieved with a minimum of 7×10-13 at τ=1000 s. The performance of this device as a function of external parameters such as light intensity, optical detuning, and cell temperature is discussed. The dependence of the dark-line resonance signal on these parameters can be understood largely by means of a simple, three-level model. The short-term stability depends critically on the optical detuning, whereas the long-term stability is limited currently by line shifts due to drifts in cell temperature.
State-of-the-art miniature atomic clocks (MACs) are based on the phenomenon of coherent population trapping (CPT) in alkali-metal atomic vapors (Rb\mathrm{Rb}Rb or Cs\mathrm{Cs}Cs). Improving the stability of the clock frequency is a crucial issue; resolving it would result in significant progress in numerous applications. Here, we examine the configuration of the light field, which is composed of two bichromatic beams with opposing handedness of circular polarization. The beams are in resonance with optical transitions in the Cs\mathrm{Cs}Cs D1{D}_{1}D1​ line ($\ensuremath{\lambda}\ensuremath{\approx}895$ nm). This configuration has previously been shown to produce observed CPT resonances with higher contrast than a standard single-beam scheme; however, our scheme implements pump and probe beams that operate independently, with the transmission of the probe beam being monitored separately. This approach differs from previous studies. The experiments are carried out using a cubic $5\ifmmode\times\else\texttimes\fi{}5\ifmmode\times\else\texttimes\fi{}5\phantom{\rule{0.2em}{0ex}}{\mathrm{mm}}^{3}$ vapor cell filled with buffer gas. It is shown that the line shape of the resonance acquires an asymmetry that can be efficiently controlled by a microwave (Raman) phase between the beams. As a proof of concept, we study the way in which this asymmetry can help to mitigate the influence of the ac Stark (light) shift on the long-term frequency stability of CPT clocks based on vapor microcells ($V\ensuremath{\ll}1\phantom{\rule{0.2em}{0ex}}{\mathrm{cm}}^{3}$). Experimental verification is performed with both a distributed-Bragg-reflector laser and a vertical-cavity surface-emitting laser. The latter is of particular importance for the development of MACs. The experimental results are in qualitative agreement with the analytical theory based on a double-$\mathrm{\ensuremath{\Lambda}}$ scheme of atomic energy levels.